```help simon2stage
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Title

Finding Simon two stage designs with extensions

Syntax

simon2stage [, options]

options               Description
-------------------------------------------------------------------------
Main
p0(#)                specifies the null proportion, the default is 0.1.
p1(#)                specifies the alternative proportion, the default
is 0.4.
alpha(#)             specifies the type I error, the default is 0.05.
beta(#)              specifies the type 2 error, the default is 0.2.
minn(#)              specifies the start sample size in the initial
grid search, the default is 1.
maxn(#)              specifies the largest sample size in the initial
grid search, the default is 35.
optimal              specifies that the interest is in the optimal
design rather than the minimax.
optp(#)              specifies the true proportion to optimize the
design for, the default is p0.
eff                  specifies that the design can stop for efficacy as
well as futility.
deltaminimax         specifies that the delta-minimax design is found.
should be found.
-------------------------------------------------------------------------

Description

The Simon two stage design is a single arm study with an interim
analysis. The main purpose of this design is to investigate whether an
intervention works or not and to stop the study early for futility.
Under the null hypothesis the probability of a success is p0, this is
usually taken as the probability of success for the current standard
treatment.  The probability of a success in this study, p, is tested
using the null hypothesis H0:p=p0 versus the alternative hypothesis
H1:p>=p1. The probability of success for the alternative hypothesis is
fixed to be a pre-specified value p1, where p1>p0.

The Simon two stage design consists of studying n1 participants in a
first stage and the study stops if there are r1 or fewer responders to
the intervention. If there are more than r1 responders in the first stage
then the study continues until n participants in total are studied. Then
the null hypothesis is not rejected if there are r or fewer responders.
Each design must satisfy the type 1 (alpha) and type 2 (beta) errors.

The probability of not rejecting H0 can be calculated conditional on any
p and let this function be R(p).  The design must therefore satisfy the
constraints R(p0)>=1-alpha and R(p1)<=beta.  The minimax design is one
that satisfies these constraints with the smallest total sample size n
and the smallest expected sample size under H0.  The alternative is to
find the "optimal" design which is the design with the smallest expected
sample size conditional on the true proportion being specified by the
optp() option.  The default is optimising the design under the null
proportion p0 and this is the classical Simon two stage design, however
this command allows greater flexibility.

The Simon two stage design has been extended here to allow for stopping
for efficacy. The design can be indexed by 5 numbers (r1 r2/n1, r/n). Now
the study stops for futility if there are r1 or fewer responders OR stops
for efficacy if there are more than r2 responders in the first stage. If
the study continues to the second stage then the null hypothesis is no
rejected if there are r or fewer responders. Again the type 1 and type 2
errors must be satisfied for a design to be considered.

There are two additional designs that are of interest firstly the
delta-minimax design as described in Shuster 2002 and secondly the set of
admissible designs as described by Mander et al. (2010). The set of
admissible designs are displayed by using a novel figure to help with
decision making.

The latest version is always kept on the SSC website. To install the

ssc install simon2stage, replace.

Options

+------+
----+ Main +-------------------------------------------------------------

p0(#) specifies the null proportion, the default is 0.1.

p1(#) specifies the alternative proportion, the default is 0.4.

alpha(#) specifies the type I error, the default is 0.05.

beta(#) specifies the type II error, the default is 0.2.

minn(#) specifies the start total sample size in the initial grid search,
the default is 1.

maxn(#) specifies the largest total sample size in the initial grid
search, the default is 35.

optimal specifies that the interest is in the optimal design rather than
the minimax.

optp(#) specifies the true proportion to optimize the design for, the
default is p0.

eff specifies that the design can stop for efficacy as well as futility.

deltaminimax specifies that the delta-minimax design is found.

Examples

simon2stage

The minimax design is {1/8 3/13}, if there are no responders or one
responder out of the first 8 participants then the study stops and the
null hypothesis is not rejected. If the study proceeds to the second
stage then the null hypothesis is rejected if there are more than 3
responders. Under the null hypothesis there is a 0.813 chance that the
study stops at the first stage and hence the average sample size is
8*0.813+13*(1-0.813) = 8.934.

simon2stage, optimal

The optimal design is {1/7 3/15}, this is similar to the previous design
but there is now a 0.85 chance that the study finishes at stage 1 and the
average sample size under the null hypothesis is 8.198 which is slightly
smaller than the minimax desgin.

simon2stage, optimal optp(0.2)

The optimal design is still {1/7 3/15} but the average sample size is
much greater at 10.386 as a direct result of a smaller chance 0.577 of
early termination.

simon2stage, optimal optp(0.2) eff

The optimal design is {(1 2)/7 3/15} the expected sample size, 9.202, is
smaller than the previous design because of a greater chance of an early
termination 0.725.

simon2stage,optimal eff deltaminimax

The delta-minimax design is {(1 2)/7 3/15} the expected sample size,
9.550, is larger than the previous designs because this is the maximum
expected sample size.

simon2stage, optp(0.2)

The minimax design optimised at a true response of 0.2 is {1/8 3/13}. The
expected sample size, 10.483, much greater than when calculating the
expected sample size at the null value. This has not altered the minimax
design.

The set of admissible designs is plotted. Setting the maximum to be 16
means that the set of admissible designs are found by fixing the maximum
sample size to be 16 or lower (this limits the designs to search
through).  Interestingly the design {(1 2)/7 3/15} is only good if there
is little weight given to the overall sample size.

Author

Adrian Mander, MRC Biostatistics Unit, Cambridge, UK.

References

R.P.A'Hern (2001) Sample size tables for exact single-stage phase II
designs. Statistics in Medicine 20:859-866.

T.R. Fleming (1982) One-sample multiple testing procedure for phase II
clinical trials. Biometrics 38:143-151.

Richard Simon (1989) Optimal two-stage designs for phase II clinical
trials. Controlled Clinical Trials 10:1-10.

A.P. Mander and S.G. Thompson (2010) Two-stage designs optimal under the
alternative hypothesis for phase II cancer trials. Contemporary Clinical
Trials 31(6):572-578.

A.P. Mander, J.M. Wason, M.J. Sweeting and S.G. Thompson (2010)
Admissible two-stage designs for phase II cancer clinical trials.
(Submitted)

Also see

Related commands

HELP FILES              SSC installation links    Description

samplesize (if installed)        (ssc install samplesize)        Sample
Size graphics
sampsi_fleming (if installed)    (ssc install sampsi_fleming)    Sample
Size for Fleming design
sampsi_reg (if installed)        (ssc install sampsi_reg)        Sample
Size for linear regression
sampsi_mcc (if installed)        (ssc install sampsi_mcc)        Sample
Size for matched case/control studies
sampsi_rho (if installed)        (ssc install sampsi_rho)        Sample
Size for Pearson correlation

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